While the grue is an enemy it is not an entity, there is no model for grue. The name comes from the code which actually references this part of the game as "grue".
The grue is a monster that will attack you if you are in total darkness for too long, you will know when you are in total darkness when your character says, "It's so dark!". After a few seconds you will hear a noise and your character will say, "What was that?" then you will take about 55 damage, and your character will say, "Ow something bit me!". Then the process repeats and you are killed by the grue unless you have a logsuit or get into light immediately. Willow is essentially immune to the grue because right before your character says, "What was that noise?" she will light a small fire restarting the time it takes for the grue to kill her.
With the Insanity update, Grue has possibly gotten a model, long dark hands, that extinguish your campfire. To prevent them from doing that, simply walk over them, they will back off, and the process wil repeat. When Grue is "attacking", music starts playing, something like a lullaby, eventually if you ignore it, it will indeed put you to rest... When playing with Maxwell he mentions "Charlie? Is that you? It's me Maxwell!"
This is a reference to the Text adventure Zork which contains Grues.